Analytical Solutions For The Forced Harmonic Vibration Of Nanobeams With Surface Effect

With the rapid development of nanotechnology,nanomaterials have been widely used in various fields.On this basis,the concept of nano-electromechanical system was proposed.Nano-electromechanical systems has great advantages in a variety of fields over the traditional electromechanical systems.The development of nano-electromechanical systems will certainly bring out revolutionary progress in these fields.However,due to the surface effect,the properties of nanostructures are size dependent,which is known as scale effect.This effect makes it difficult to exactly describe the mechanical performance of these nanostructures.In order to promote the development of nano-electromechanical systems,it is necessary to develop mechanics theory about nanostructures.Nanobeam is the most commonly used structure in nano-electromechanical system.The study on the vibration characteristics of nano-beam is one of the important parts of the mechanices for nano-structures.In view of the current progress of the studies on nano-beam,the following work is completed and presented in this paper:(1)With the help of the governing equations based on Gurtin-Murdoch surface elasticity model,the effect of surface energy on the forced harmonic vibration of nanobeams is studied with analytical method.Timoshenko beam model and Euler-Bernoulli beam model are considered,and the corresponding analytical solutions are obtained.In this process,Green's function method is adopted,and the exact and closed-form Green's functions for simply supported beam,clamped beam and cantilever beam are derived with separation of variables and Laplace transformation.From numerical results,it is observed that surface energy has a significant influence on the stiffness of beam,and the influence is related with boundary conditions and size of the beam.(2)The rotational spring model for crack in beam is introduced in the analysis of cracked nanobeam.In the frame work of Euler-Bernoulli beam model,Green's function for a nanobeam with one vertical crack under harmonic force is investigated.The solutions for simply supported beam,clamped beam and cantilever beam are obtained.From numerical results,it is found that the stiffness of the beam is related with parameters such as crack length and crack location.